Computational Arithmetic Geometry

TL;DR: In this article, the average-case complexity of Diophantine problems over C, Q, and Z/pZ was studied, and it was shown that for almost all inputs, the problem can be solved in polynomial time within coNP.

Abstract: We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I.Given a polynomial f?Zv, x, y, decide the sentence ?v?x?yf(v, x, y)=?0, with all three quantifiers ranging over N (or Z). II.Given polynomials f1, ?, fm?Zx1, ?, xn with m?n, decide if there is a rational solution to f1=?=fm=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PNPNP, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems, which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of cryptosystems.